Use the 20 sequences below to derive general formulae for t4 through to t12 for any other sequence in the same pattern given t1, t2 and t3

[timothyjohnson]

tn is defined as: 0 > tn ≤ 2048

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12

Sequence 1

1188​

861​

1608​

205​

788​

370​

2040​

1907​

411​

143​

276​

328​

Sequence 2

302​

1667​

615​

238​

51​

510​

2044​

40​

205​

293​

533​

1337​

Sequence 3

197​

1725​

1866​

1158​

1536​

1905​

723​

1336​

412​

1554​

1463​

924​

Sequence 4

101​

352​

262​

1683​

22​

644​

346​

1843​

1838​

577​

1946​

1635​

Sequence 5

424​

1586​

1210​

1469​

563​

610​

1424​

69​

173​

638​

574​

1217​

Sequence 6

1263​

1030​

688​

748​

2005​

1554​

430​

175​

417​

510​

1747​

1482​

Sequence 7

1979​

156​

1338​

1004​

1433​

672​

555​

1451​

1703​

452​

1171​

815​

Sequence 8

1179​

98​

1279​

601​

1224​

446​

905​

549​

1972​

651​

1161​

1261​

Sequence 9

1157​

1326​

1655​

380​

529​

1477​

537​

932​

509​

1511​

1785​

893​

Sequence 10

1160​

1152​

813​

1064​

1171​

48​

1894​

322​

1926​

1074​

1857​

755​

Sequence 11

1130​

1746​

1009​

1609​

752​

6​

713​

1277​

1253​

479​

1006​

308​

Sequence 12

116​

1647​

1944​

1156​

2016​

2028​

853​

1561​

1967​

760​

1040​

1712​

Sequence 13

208​

699​

942​

1171​

360​

1701​

327​

856​

330​

14​

1844​

686​

Sequence 14

886​

8​

1928​

748​

1128​

246​

286​

502​

172​

1825​

1083​

708​

Sequence 15

1719​

2005​

419​

1613​

1095​

1926​

828​

227​

1757​

1485​

782​

1488​

Sequence 16

1482​

1305​

1068​

893​

1018​

1732​

1328​

1246​

1305​

617​

1178​

1654​

Sequence 17

1529​

363​

786​

624​

1476​

1242​

547​

1453​

41​

56​

1285​

1633​

Sequence 18

1832​

892​

541​

972​

1208​

1287​

1208​

1825​

1144​

1213​

704​

1231​

Sequence 19

1772​

1434​

1208​

1829​

556​

1047​

1205​

173​

1225​

391​

503​

1209​

Sequence 20

1273​

376​

1431​

1545​

36​

1869​

1255​

386​

865​

472​

154​

36​

 

[blamocur]

Any chance you could post those 240 values as plain text or a CSV file?
Thank you.

 

[topsquark]

tn is defined as: 0 > tn ≤ 2048

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12

Sequence 1

1188​

861​

1608​

205​

788​

370​

2040​

1907​

411​

143​

276​

328​

Sequence 2

302​

1667​

615​

238​

51​

510​

2044​

40​

205​

293​

533​

1337​

Sequence 3

197​

1725​

1866​

1158​

1536​

1905​

723​

1336​

412​

1554​

1463​

924​

Sequence 4

101​

352​

262​

1683​

22​

644​

346​

1843​

1838​

577​

1946​

1635​

Sequence 5

424​

1586​

1210​

1469​

563​

610​

1424​

69​

173​

638​

574​

1217​

Sequence 6

1263​

1030​

688​

748​

2005​

1554​

430​

175​

417​

510​

1747​

1482​

Sequence 7

1979​

156​

1338​

1004​

1433​

672​

555​

1451​

1703​

452​

1171​

815​

Sequence 8

1179​

98​

1279​

601​

1224​

446​

905​

549​

1972​

651​

1161​

1261​

Sequence 9

1157​

1326​

1655​

380​

529​

1477​

537​

932​

509​

1511​

1785​

893​

Sequence 10

1160​

1152​

813​

1064​

1171​

48​

1894​

322​

1926​

1074​

1857​

755​

Sequence 11

1130​

1746​

1009​

1609​

752​

6​

713​

1277​

1253​

479​

1006​

308​

Sequence 12

116​

1647​

1944​

1156​

2016​

2028​

853​

1561​

1967​

760​

1040​

1712​

Sequence 13

208​

699​

942​

1171​

360​

1701​

327​

856​

330​

14​

1844​

686​

Sequence 14

886​

8​

1928​

748​

1128​

246​

286​

502​

172​

1825​

1083​

708​

Sequence 15

1719​

2005​

419​

1613​

1095​

1926​

828​

227​

1757​

1485​

782​

1488​

Sequence 16

1482​

1305​

1068​

893​

1018​

1732​

1328​

1246​

1305​

617​

1178​

1654​

Sequence 17

1529​

363​

786​

624​

1476​

1242​

547​

1453​

41​

56​

1285​

1633​

Sequence 18

1832​

892​

541​

972​

1208​

1287​

1208​

1825​

1144​

1213​

704​

1231​

Sequence 19

1772​

1434​

1208​

1829​

556​

1047​

1205​

173​

1225​

391​

503​

1209​

Sequence 20

1273​

376​

1431​

1545​

36​

1869​

1255​

386​

865​

472​

154​

36​

It would help to know what you've been doing with sequences. Are you looking at arithmetic and geometric sequences? Other types?

-Dan

 

[Steven G]

tn is defined as: 0 > tn ≤ 2048
This means that 0>tn (so tn<0, ie tn is negative) AND tn ≤ 2048.
Combining yields that tn<0.
Is that what you want?

 

[blamocur]

Any chance you could post those 240 values as plain text or a CSV file?
Thank you.

Never mind, I've extracted it from your PDF. Here it is in case someone wants it too:

1188 861 1608 205 788 370 2040 1907 411 143 276 328 302 1667 615 238 51 510 2044 40
205 293 533 1337 197 1725 1866 1158 1536 1905 723 1336 412 1554 1463 924 101 352 262 1683
22 644 346 1843 1838 577 1946 1635 424 1586 1210 1469 563 610 1424 69 173 638 574 1217
1263 1030 688 748 2005 1554 430 175 417 510 1747 1482 1979 156 1338 1004 1433 672 555 1451
1703 452 1171 815 1179 98 1279 601 1224 446 905 549 1972 651 1161 1261 1157 1326 1655 380
529 1477 537 932 509 1511 1785 893 1160 1152 813 1064 1171 48 1894 322 1926 1074 1857 755
1130 1746 1009 1609 752 6 713 1277 1253 479 1006 308 116 1647 1944 1156 2016 2028 853 1561
1967 760 1040 1712 208 699 942 1171 360 1701 327 856 330 14 1844 686 886 8 1928 748
1128 246 286 502 172 1825 1083 708 1719 2005 419 1613 1095 1926 828 227 1757 1485 782 1488
1482 1305 1068 893 1018 1732 1328 1246 1305 617 1178 1654 1529 363 786 624 1476 1242 547 1453
41 56 1285 1633 1832 892 541 972 1208 1287 1208 1825 1144 1213 704 1231 1772 1434 1208 1829
556 1047 1205 173 1225 391 503 1209 1273 376 1431 1545 36 1869 1255 386 865 472 154 36

 

[timothyjohnson]

tn is defined as: 0 > tn ≤ 2048
This means that 0>tn (so tn<0, ie tn is negative) AND tn ≤ 2048.
Combining yields that tn<0.
Is that what you want?

Hi, thanks for the reply sir. I meant tn is positive values less than or equal to 2048.

 

[timothyjohnson]

Never mind, I've extracted it from your PDF. Here it is in case someone wants it too:

1188 861 1608 205 788 370 2040 1907 411 143 276 328 302 1667 615 238 51 510 2044 40
205 293 533 1337 197 1725 1866 1158 1536 1905 723 1336 412 1554 1463 924 101 352 262 1683
22 644 346 1843 1838 577 1946 1635 424 1586 1210 1469 563 610 1424 69 173 638 574 1217
1263 1030 688 748 2005 1554 430 175 417 510 1747 1482 1979 156 1338 1004 1433 672 555 1451
1703 452 1171 815 1179 98 1279 601 1224 446 905 549 1972 651 1161 1261 1157 1326 1655 380
529 1477 537 932 509 1511 1785 893 1160 1152 813 1064 1171 48 1894 322 1926 1074 1857 755
1130 1746 1009 1609 752 6 713 1277 1253 479 1006 308 116 1647 1944 1156 2016 2028 853 1561
1967 760 1040 1712 208 699 942 1171 360 1701 327 856 330 14 1844 686 886 8 1928 748
1128 246 286 502 172 1825 1083 708 1719 2005 419 1613 1095 1926 828 227 1757 1485 782 1488
1482 1305 1068 893 1018 1732 1328 1246 1305 617 1178 1654 1529 363 786 624 1476 1242 547 1453
41 56 1285 1633 1832 892 541 972 1208 1287 1208 1825 1144 1213 704 1231 1772 1434 1208 1829
556 1047 1205 173 1225 391 503 1209 1273 376 1431 1545 36 1869 1255 386 865 472 154 36

Thanks for the reply and thanks for the effort too. Every reply gets me closer to the answer.

 

[timothyjohnson]

It would help to know what you've been doing with sequences. Are you looking at arithmetic and geometric sequences? Other types?

-Dan

I tried using the formula for AP, GP and even Fibonacci but they didn't work but I did notice a lot of patterns that indicate the tn is derived from a combination of the previous value, n and maybe another external value. For instance in Sequence 20: t12 and t5 are both 36 and are preceded by similar values 154.. This might indicate a relationship between the formular for both t5 and t12 have just slight differences. This kind of pattenr can be seen all over the sequences but I don't know of a formula that can help to derive this. Every reply stirs me in the right direction. Thanks a lot for the help.

 

[timothyjohnson]

Observations

Observation 1:

It seemed to me as if for each sequence to be distinct, 2 values x & y were introduced such that:

For Sequence 1, x = 1 and y =210 to give the formulae:

t2 = (2048 – t1) + x

t3 = (2048 + t2) + 2*y – x

t4 = y - 5

t5 = (2048 – t4) - 5*y

t6 = (2048 – t5) - (4*y + 46)

t7 = y*x + 145

t8 = (2048 + t7) + 9*y

t9 = 2*y + 11

t10 = (2048 – t9) - (7*y + 17)

t11 = (2048 – t10) - (8*y - 59)

t12 = t1 - t2 + x

However, these are only true for Sequence 1.



Observation 2:

Observation for sequence 1​
n	2048​		Tn-1​	​	Value​	​	T​	Multiples of 211​	Multiples of 211​	Remainder​
1	​	​		​	​	​	1188​	​		​
2	2048​	-​	1188​	+​	1​	=​	861​	0​	n - 2	1​
3	2048​	-​	861​	+​	421​	=​	1608​	2​	n - 1	-1​
4	2048​	-​	1608​	+​	-235​	=​	205​	-1​	n - 5	187​
5	2048​	-​	205​	+​	-1055​	=​	788​	-5​	n - 10	0​
6	2048​	-​	788​	+​	-890​	=​	370​	-4​	n - 10	165​
7	2048​	-​	370​	+​	362​	=​	2040​	1​	n - 6	151​
8	2048​	-​	2040​	+​	1899​	=​	1907​	9​	n + 1	0​
9	2048​	-​	1907​	+​	270​	=​	411​	1​	n - 8	59​
10	2048​	-​	411​	+​	-1494​	=​	143​	-7​	n - 17	194​
11	2048​	-​	143​	+​	-1629​	=​	276​	-7​	n - 18	59​
12	2048​	-​	276​	+​	-1444​	=​	328​	-6​	n - 18	33​

Other notable observations:

1. Sequence 20: t12 and t5 are both 36 and are preceded by similar values 154.. This might indicate a

It would help to know what you've been doing with sequences. Are you looking at arithmetic and geometric sequences? Other types?

-Dan

relationship between the formula for both t5 and t12 have just slight differences.

 

[Steven G]

Hi, thanks for the reply sir. I meant tn is positive values less than or equal to 2048.

That is written as 0<tn≤ 2048

 

[timothyjohnson]

That is written as 0<tn≤ 2048

Yes, you're correct. It was a mistake.

 

[blamocur]

1. Sequence 20: t12 and t5 are both 36 and are preceded by similar values 154.. This might indicate a

But in Seq.20 t4 = 1545, not 154.

 

[blamocur]

It seemed to me as if for each sequence to be distinct, 2 values x & y were introduced such that:

If I understood correctly the text of the attachment then t1,t2,t3 can be chosen arbitrarily and the rest (t4,...,t12) can be derived from them. This would mean that your 'x' and 'y' would have to be derived from t1,t2 and t3.

All I can see at this point is that the dependencies are not linear because the rank the whole matrix is 12, but for t4,...t12 to be linearly dependent with t1,t2,t3 the rank would have to be 3.

Can you give us some background here? I.e. where does this problem come from? If it is a part of your homework then what topic is it related to?

 

[timothyjohnson]

Hi everyone. Thanks for your comments and support. I was able to find a websites online that hosts sequences. In it, I found a specific one that's related to the first sequence: https://oeis.org/A075232. I found out that most of the values in my first sequence, have n added to them and are listed there. The issue now is, I don't know what "Numbers k such that k^9 is an interprime = average of two successive primes." really means. Any help will be greatly appreciated.